Comparison between images via bilinear fuzzy relation equations

We present a comparison between two images A and B based on the greatest solution of a system of bilinear fuzzy relation equations A○x = B○x, where “○” is the max–min composition, being A and B known as fuzzy relations and x is unknown. Here A and B are images considered as fuzzy relations being their pixels normalized in [0, 1] with respect to the grey scale used. Due to symmetry of every equation involved, A (resp., B) could be the original image and B (resp., A) is an image modified of A (resp., B), for instance, either noised or watermarked. The comparison is made by using an index which is more robust than other two indices used in previous works: the first one is based on the greatest eigen fuzzy set (with respect to max–min composition) and smallest eigen fuzzy set (with respect to min–max composition) and the second one is based on the Lukasiewicz triangular norm. The comparison is made between the original image and the same image with noise introduced at several values σ of the standard deviation.